Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420064 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2572 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
881 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6)(7,9)(8,16,18,12)(10,13)(11,14)(15,17,20,19);(5,7,9,14,11,13,10,6)(8,12,17,15,18,16,19,20);(5,8,9,17,11,18,10,19)(6,12,7,15,14,16,13,20);(2,4)(6,7,14,13)(12,15,16,20)Polycyclic presentation:
2^9|a^2=gh,b^2=ef,c^2=ef,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72227424 |
| maximal normal subgroup of | #72227424 |
| maximal quotient of | #72268632 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 359651);